Answer
$-\dfrac{x + 4}{x - 5}$
Restriction: $x \ne 3, 5$
Work Step by Step
To solve these types of problems, we want to remove greatest common factors in both the numerator and denominator. Let us first try to factor the numerator and denominator:
Let's factor out a $-1$ out of $12 - x - x^2$ so we can rewrite it with the greatest degree term first:
$(-1)(x^2 + x - 12)$
Now, we can factor:
$(-1)(x^2 + x - 12)$ = $-1(x + 4)(x - 3)$
$x^2 - 8x + 15$ = $(x - 5)(x - 3)$
Let's plug the factors back into the expression:
$\dfrac{-1(x + 4)(x - 3)}{(x - 5)(x - 3)}$
We can factor out $x - 3$ from the numerator and denominator. Move the $-1$ outside the fraction:
$-\dfrac{x + 4}{x - 5}$
To see what restrictions for the variable we have, we need to see which values for the variable will make the denominator equal to $0$, which will cause the fraction to become undefined. Let's set the denominator equal to $0$ to see which values for the variable we cannot use:
$(x - 5)(x - 3) = 0$
Set each factor each to $0$:
First factor:
$x - 5 = 0$
Add $5$ to each side of the equation:
$x = 5$
Second factor:
$x - 3 = 0$
Add $3$ to both sides of the equation to isolate the $x$ term:
$x = 3$
Restriction: $x \ne 3, 5$
